Low-frequency–high-intensity limit of the Keldysh-Faisal-Reiss theory

Abstract

When a frequency of the circularly polarized laser field approaches zero the above threshold ionization rate should approach the well-known static-field limit of tunneling ionization. In the high-intensity limit of the laser field the Keldysh-Faisal-Reiss (KFR) theory is expected to be valid. For the ground state of a hydrogen atom we study various forms of the KFR theory when both conditions: $\ensuremath{\omega}⪡1\phantom{\rule{0.3em}{0ex}}\mathrm{a}.\mathrm{u}.$ and $\ensuremath{\gamma}⪡1\phantom{\rule{0.3em}{0ex}}(\ensuremath{\omega}$ is the frequency and $\ensuremath{\gamma}$ the Keldysh parameter) are satisfied. For the circularly polarized laser field ionization rate in the Keldysh theory [which utilizes the length gauge $(\stackrel{P\vec}{d}∙\stackrel{P\vec}{E})$ form of the matrix element] is calculated analytically. We show numerically that if the WKB Coulomb correction in the final state of the ionized electron is included, the Keldysh theory gives the correct result in the tunneling domain. In the barrier-suppression regime the Keldysh theory without this correction gives ionization rates close to the exact static-field results. The Reiss theory [which utilizes the velocity gauge $(\stackrel{P\vec}{p}∙\stackrel{P\vec}{A})$ form of the matrix element] leads to too small ionization rates in the limit $\ensuremath{\omega}\ensuremath{\rightarrow}0$, $\ensuremath{\gamma}\ensuremath{\rightarrow}0$.